EM 1110-2-1100 (Part II)
30 Apr 02
(e) In deep and intermediate water depths, the significant wave height obtained by the spectral analysis
using the above equation is usually greater than that from the wave train analysis. The zero-crossing period
from the spectral method is only an approximation, while the period associated with the largest wave energy
known as the peak period Tp, can only be obtained via the spectral analysis. In the spectral representation
of swell waves, there is a single value of the peak period and wave energy decays at frequencies to either side.
The spectra for storm waves is sometimes multi-peaked. One peak (not always the highest) corresponds to
the swell occurring at lower frequencies. One and sometimes more peaks are associated with storm waves
occurring at comparatively higher frequencies. In a double-peaked spectra for storm waves, the zero-crossing
period generally occurs at higher frequencies than the peak period. In a multi-peaked spectrum, the zero-
crossing period is not a measure of the frequency where peak energy occurs.
(5) Relationships among H1/3, Hs, and Hm0 in shallow water.
(a) By conception, significant height is the average height of the third-highest waves in a record of time
period. By tradition, wave height is defined as the distance from crest to trough. Significant wave height Hs
can be estimated from a wave-by-wave analysis in which case it is denoted H1/3, but more often is estimated
from the variance of the record or the integral of the variance in the spectrum in which case it is denoted Hm0.
Therefore, Hs in Equation II-1-152 should be replaced with Hm0 when the latter definition of Hs is implied.
While H1/3 is a direct measure of Hs, Hm0 is only an estimate of the significant wave height which under many
circumstances is accurate. In general in deep water H1/3 and Hm0 are very close in value and are both
considered good estimates of Hs. All modern wave forecast models predict Hm0 and the standard output of
most wave gauge records is Hm0. Few routine field gauging programs actually compute and report H1/3 and
report as Hs with no indication of how it was derived. Where H1/3 and Hs are equivalent, this is of little
(b) Thompson and Vincent (1985) investigated how H1/3 and Hm0 vary in very shallow water near
breaking. They found that the ratio H1/3/Hm0 varied systematically across the surf zone, approaching a
maximum near breaking. Thompson and Vincent displayed the results in terms of a nomogram (Figure II-1-
40). For steep waves, H1/3/Hm0 increased from 1 to about 1.1, then decreased to less than 1 after breaking.
For low steepness waves, the ratio increased from 1 before breaking to as much as 1.3-1.4 at breaking, then
decreased afterwards. Thompson and Vincent explained this systematic variation in the following way. As
low steepness waves shoal prior to breaking, the wave shape systematically changes from being near
sinusoidal to a wave shape that has a very flat trough with a very pronounced crest. Although the shape of
the wave is significantly different from the sine wave in shallow water, the variance of the surface elevation
is about the same, it is just arranged over the wave length differently from a sine wave. After breaking, the
wave is more like a bore, and as a result the H1/3 can be smaller (by about 10 percent) than Hm0.
(c) The critical importance of this research is in interpreting wave data near the surf zone. It is of
fundamental importance for the engineer to understand what estimate of significant height he is using and
what estimate is needed. As an example, if the data from a gauge is actually Hm0 and the waves are near
breaking, the proper estimate of Hs is given by H1/3. Given the steepness and relative depth, H1/3 may be
estimated from Hm0 by Figure II-1-40. Numerically modelled waves near the surf zone are frequently
equivalent to Hm0. In this case, Hs will be closer to H1/3 and the nomogram should be used to estimate Hs.
(6) Parametric spectrum models.
(a) In general, the spectrum of the sea surface does not follow any specific mathematical form.
However, under certain wind conditions the spectrum does have a specific shape. A series of empirical
expressions have been found which can be fit to the spectrum of the sea surface elevation. These are called
parametric spectrum models, and are useful for routine engineering applications. A brief description of these
Water Wave Mechanics